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Properties

Label 1-312-312.179-r0-0-0
Degree $1$
Conductor $312$
Sign $0.0128 + 0.999i$
Analytic cond. $1.44892$
Root an. cond. $1.44892$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s + 25-s + (−0.5 + 0.866i)29-s + 31-s + (0.5 + 0.866i)35-s + (−0.5 + 0.866i)37-s + (−0.5 + 0.866i)41-s + (−0.5 − 0.866i)43-s − 47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  − 5-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s + 25-s + (−0.5 + 0.866i)29-s + 31-s + (0.5 + 0.866i)35-s + (−0.5 + 0.866i)37-s + (−0.5 + 0.866i)41-s + (−0.5 − 0.866i)43-s − 47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.0128 + 0.999i$
Analytic conductor: \(1.44892\)
Root analytic conductor: \(1.44892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 312,\ (0:\ ),\ 0.0128 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4713503515 + 0.4653443347i\)
\(L(\frac12)\) \(\approx\) \(0.4713503515 + 0.4653443347i\)
\(L(1)\) \(\approx\) \(0.7465817459 + 0.1206219563i\)
\(L(1)\) \(\approx\) \(0.7465817459 + 0.1206219563i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 - T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.74158688359579583540068778300, −24.29094366378141668050808474219, −23.09753799987990658965828471949, −22.481842487458425511492453708581, −21.46764119070384463706541015681, −20.507662416815826173642862155905, −19.443992461067245567567781099456, −18.82234121131917940028036123781, −18.03351797013672812074408491139, −16.53106126856416400477538506872, −15.90226761848129098876483692004, −15.21496755494359392444228079026, −14.016875864183851654803256653445, −12.957899572371541884778266407386, −11.94005843768958972358138088828, −11.33747558623304943651762114597, −10.07182558115560567850438931126, −8.91181196128580290444300861922, −8.11083552523309312365999782937, −7.024905906932659837910175084895, −5.8368694219824291213300566338, −4.7765550947178850973316340201, −3.41212603194673413398725914035, −2.576884466130434342975863310075, −0.44852953891761525150470755134, 1.437134956032961238910442804674, 3.24716398574506116369336191232, 4.001431328363725719130289688429, 5.17504499941982169629115734483, 6.63603443900669973330624955994, 7.5531611525091141371549982079, 8.27062433121248901952749068771, 9.824991233604486508519915139962, 10.44580700842380430045711514997, 11.68511020971729805684349501044, 12.50722467518767748116718060340, 13.44475539367392764971128035463, 14.61549675249270728336791249811, 15.4832887547734464348537332532, 16.34903154852307590856974511782, 17.18524329802097242133951483167, 18.35599372007455182792349668870, 19.27864193070516170397465341778, 20.075379521656384483875541726883, 20.7212298543821189082413545176, 22.03014311169875277316072609189, 23.112066112364509330070022032144, 23.38852292539175092224481694700, 24.3942082349255578348661894444, 25.69672524574537568776624138152

Graph of the $Z$-function along the critical line